How would this work?
The packages that could use it would still need to ship a configuration
file for every web server since there is no common API like
/usr/lib/cgi-bin/ .

You're right, but a FastCGI server needs a webserver as frontend which
can expose it. My idea was to use this virtual package for webserver
able to establish the link between a browser and a FastCGI server
(Nginx, libapache2-mod-fcgid,...)